Isolated singularities of solutions of defocusing Hartree equation. (June 2017) - Wang, Ying
Abstract: Our purpose of this paper is to study isolated singular positive solution u of defocusing Hartree equation (1) { − Δ u + u + I α [ u p ] u q = 0 in R N ∖ { 0 }, lim | x | → + ∞ u ( x ) = 0, where p, q > 0, N ≥ 3, α ∈ ( 0, N ) and I α [ u p ] ( x ) = ∫ R N u ( y ) p | x − y | N − α d y . We obtain that the solution u ∈ L l o c 1 ( R N ) of(1) satisfying that u p ∈ L 1 ( R N ), I α [ u p ] u q ∈ L l o c 1 ( R N ), is a weak solution of (2) { − Δ u + u + I α [ u p ] u q = k δ 0 in R N, lim | x | → + ∞ u ( x ) = 0 . Furthermore, the classical solution of(1) is derived by considering the very weak solution of(2) . To this end, we make use of Schauder fixed point theorem to obtain the existence of weak solutions of(2) .